Algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere and finite braid group orbits on character varieties

Abstract

We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\ge 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such a connection. We relate this property to a peculiarity of the corresponding monodromy representation: to yield a finite braid group orbit on the appropriate character variety. Under reasonable assumptions on the deformed connection, we may actually establish an equivalence between both properties. We apply this result in the rank two case to relate finite branching and algebraicity for solutions of Garnier systems. For general rank, a byproduct of this work is a tool to produce regular flat meromorphic connections on vector bundles over projective varieties of high dimension.

Appendix

We want to explain here the generalization of Lemma 4.4.2 of [23] we need to obtain Theorem 9 with our slightly weakened hypotheses. The generalized lemma is as follows. In the sequel we use the terminology of [23], beware that the wording “logarithmic pole” refers to the form of the local solution of the considered Fuchsian scalar equation.

Lemma 9

Take an integrable system

$$\begin{aligned} dY=\left( \left[ {\begin{matrix} 0&{}1\\ p(z,t)&{}0\end{matrix}} \right] dz+\Omega \right) \cdot Y \end{aligned}$$

where $\Omega =\sum _{i=1}^N\Omega _i(z,t) dt_i$ and p is given by Eq. (6). Suppose $z\mapsto p(z,t^0)$ has exactly $2N+2$ poles in $\mathbb {C}$ (without multiplicity) and the monodromy of the system around any of the $N+3$ poles $z=0,1,\infty$ and $z=t_i,i=1,\ldots ,N$ is not scalar (non-apparent poles). Then the (1, 2) entry $A_i$ of $\Omega _i$ satisfies the following.

The rational function $z\mapsto A_i(z,t^0)$

• 1. is holomorphic outside $\lambda _1,\ldots ,\lambda _N,\infty$,

• 2. has only simple poles in $\mathbb {C}\cup \{\infty \}$ and

• 3. has zeroes in the points 0, 1 and $t_j^0$, for $1\le j \le N$, $j\ne i$.

Proof

The proof is given in [23] with the hypothesis that none of the poles $z=0,1,\infty ,t_1,\ldots ,t_N$ has integer local exponent $\theta$. It is obtained through local studies in the points $z=0,1,\infty ,t_i,\lambda _i(t)$ involving local solutions of $\frac{d^2y}{dz^2}=p(z,t)y$ given by Frobenius’s method.

To complete the proof, we only need to do the analogous study at those elements of $\{0,1,\infty ,t_1,\ldots ,t_N\}$ that are logarithmic poles.

Suppose $z=t_j$ is a logarithmic pole of $\frac{d^2y}{dz^2}=p(z,t)y$, with $t_j\ne t_i$, $t_j\ne \infty$.

By the arguments of [23], equation 4.4.5 p. 185, we know

$$\begin{aligned} A_i(z,t)=u(t)g(z,t)+a(t)v_1^2+b(t)v_1v_2+c(t)v_2^2 \end{aligned}$$

(9)

for

• $v_1(z,t),v_2(z,t)$ any fundamental system of solutions of $\frac{d^2y}{dz^2}=p(z,t)y$ near $z=t_j$,

• $g(z,t):=v_1\frac{\partial }{\partial t_i}v_2-v_2\frac{\partial }{\partial t_i}v_1$

• u(t), a(t), b(t), c(t) holomorphic functions of t, with u nowhere vanishing.

By Frobenius’s method (with parameter) we may take

• $v_1=(z-t_j)^{\frac{1}{2}}h_1(z,t),$ $v_2=(z-t_j)^{\frac{1}{2}}(h_1(z,t)\log (z-t_j)+h_2(z,t))$,    if $\theta _j=0$;

• $v_1=(z-t_j)^{\frac{1+m}{2}}h_1(z,t)$, $v_2=(z-t_j)^{\frac{1-m}{2}}h_1(z,t)+k(t)(v_1 \log (z-t_j)+(z-t_j)^{\frac{1-m}{2}}h_3(z,t))$, if $\vert \theta _j \vert =m>0$.

Where the functions $h_1,h_2,h_3$ are holomorphic in the neighborhood of $z=t_j$, $h_1$ does not vanish and k(t) is a nowhere vanishing holomorphic function of t.

In any case, the right hand side of (9) takes the form

$$\begin{aligned} \sum _{\ell =0}^2 k_{\ell }(z,t)\left( \log (z-t_j)\right) ^{\ell }, \end{aligned}$$

with $k_{\ell }(z,t)$ holomorphic in the neighborhood of $z=t_j$. The lemma below and uniformity of $A_i$ allow to conclude $k_2=k_1=0$. In any case, the specific form of $k_0$ and the conditions given by $k_2=k_1=0$ allow to see $k_0(z,t)$ vanishes at $z=t_j$.

The same arguments allow to prove holomorphicity of $A_i$ at $t_i$, in case it is a logarithmic pole. If $\infty$ is a logarithmic pole of $\frac{d^2y}{dz^2}=p(z,t)y$, it remains to see that $A_i$ has at most a pole of order one at this point. The proof is almost as above, the only difference is a slight change in the form of the local fundamental system of solutions. $\square$

Lemma 10

Let $(k_{\ell }(z))_{0\le \ell \le r}$ be holomorphic functions in the punctured disc $\mathbb {D}^*:=\{z\in \mathbb {C}^*,\vert z \vert <1\}$. Consider the function

$$\begin{aligned} f(z)=\sum _{\ell =0}^r k_{\ell }(z)\left( \log z\right) ^{\ell } \end{aligned}$$

defined on the universal cover of $\mathbb {D}^*$. If f is uniform then $k_{\ell } \equiv 0$, $\ell >0$.

Proof

We proceed by induction on r. The result is trivial for $r=0$. Take $r>0$ and suppose the lemma is true for $r-1$. The monodromy group of $\log z$ is generated by $\log z \mapsto \log z+2i\pi$. Hence, f is uniform if and only if the function $g(z)=f(z)-\sum _{\ell =0}^r k_{\ell }(z)\left( \log z +2i\pi \right) ^{\ell }$ is zero. This function g(z) is obviously uniform and can be written in the form $g(z)=\sum _{\ell =0}^{r-1} q_{\ell }(z)\left( \log z \right) ^{\ell }$ with $q_{\ell }$ holomorphic in $\mathbb {D}^*$. By the induction hypothesis $2i\pi rk_r=q_{r-1}\equiv 0$. Again by the induction hypothesis, $k_{\ell }\equiv 0$ for $\ell =1,\ldots , r-1$. $\square$